Existence of good moduli spaces for A_k-stable curves

TL;DR

This paper establishes a criterion for algebraic stacks to have good moduli spaces and applies it to moduli stacks of A_k-stable curves, advancing the understanding of their geometric properties.

Contribution

It provides a general criterion for the existence of good moduli spaces and applies it to A_k-stable curves, extending the theory of moduli spaces.

Findings

Proved a criterion for good moduli spaces of algebraic stacks.

Showed that moduli stacks of A_k and A_k^+-stable curves admit good moduli spaces.

Laid groundwork for future work on projectivity and minimal model program applications.

Abstract

We prove a general criterion for an algebraic stack to admit a good moduli space. This result may be considered as a weak analog of the Keel-Mori theorem, which guarantees the existence of a coarse moduli space for a separated Deligne-Mumford stack. We apply our result to prove that the moduli stacks of A_k and A_k^+-stable curves admit good moduli spaces. In forthcoming work, we will prove that these moduli spaces are projective and use them to construct the second flip in the log minimal model program for M_g.